{"paper":{"title":"The typical structure of sparse $K_{r+1}$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'ozsef Balogh, Lutz Warnke, Robert Morris, Wojciech Samotij","submitted_at":"2013-07-23T07:48:51Z","abstract_excerpt":"Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erd\\H{o}s and R\\'enyi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph $H$. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical $H$-free graph with $n$ vertices and $m$ edges changes as $m$ grows from $0$ to $\\text{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5967","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}